In Part Two we looked at the claimed relationship ED=AA in Miskolczi’s 2007 paper.
- Ed = downward atmospheric radiation absorbed by the surface
- Aa = surface radiation absorbed in the atmosphere
I showed that they could not be exactly equal. Ferenc Miskolczi himself has just joined the discussion and confirmed:
I think I was the first who showed the AA≈ED relationship with reasonable quantitative accuracy.
That is, there is not a theoretical basis for equating AA=ED as an identity.
There is a world of difference between demonstrating a thermodynamic identity and an approximate experimental relationship. In the latter situation, it is customary to make some assessment of how close the values are and the determining factors in the relationship.
But in reviewing the 2007 paper again I noticed something very interesting:
Now the point I made in Part Two was that AA ≠ ED because the atmosphere is a little bit cooler than the surface – at the average height of emission of the atmosphere. So we would expect ED to be a a little less than AA.
Please review the full explanation in Part Two to understand this point.
Now take a look at the graph above. The straight line drawn on is the relationship ED=AA.
The black circles are for an assumption that the surface emissivity, εG = 1. (This is reasonably close to the actual emissivity of the surface, which varies with surface type. The oceans, for example, have an emissivity around 0.96).
In these calculated results you can see that Downwards Emittance, ED is a little less than AA. In fact, it looks to be about 5% less on average. (And note that is ED = Absorbed Downwards Emittance)
Of course in practice, εG < 1. What happens then?
Well, in the graph above, with εG = 0.96 the points appear to lie very close to the line of ED=AA.
I think there is a calculation error in Miskolczi’s paper – and if this is true it is quite fundamental. Let me explain..
Here is the graphic for explaining Miskolczi’s terms:
When the surface is a blackbody (εG =1), SU = SG – that is, the upwards radiation from the surface = the emitted radiation from the ground.
The terms and equations in his 2007 are derived with reference to the surface emitting as a blackbody.
When εG < 1, some care is needed in rewriting the equations. It looks like this care has not been taken and the open circles in his Fig 2 (my figure 1) closely matching the ED=AA line are an artifact of incorrectly rewriting the equations when εG < 1.
That’s how it looks anyway.
Here is my graphic for the terms needed for this problem:
As much as possible I have reused Miskolczi’s terms. Because the surface is not a blackbody, the downward radiation emitted by the atmosphere is not completely absorbed. So I created the term EDA for the emission of radiation by the atmosphere. Then some of this, Er, is reflected and added to SG to create the total upward surface radiation, SU.
Note as well that the relationship emissivity = absorptivity is only true for the same wavelengths. See note 4 in Part Two.
Now for some necessary maths – it is very simple. All we are doing is balancing energy to calculate the two terms we need. (Updated note – some of the equations are approximations – the real equation for emission of radiation is a complex term needing all of the data, code and a powerful computer – but the approximate result should indicate that there is an issue in the paper that needs addressing – see comment).
And the objective is to get a formula for the ratio ED/AA – if ED=AA, this ratio = 1. And remember that in Figure 1, the relationship ED/AA=1 is shown as the straight line.
First, instead of having the term for atmospheric temperature, let’s replace it with:
TA = TS – ΔT 
where ΔT represents the idea of a small change in temperature.
Second, the emitted atmospheric downward radiation comes from the Stefan-Boltzmann law:
EDA = εAσ(TS – ΔT)4 
Third, downward atmospheric radiation absorbed by the surface:
ED = εGEDA 
Fourth, the upward surface radiation is the emitted radiation plus the reflected atmospheric radiation. Emitted radiation is from the Stefan-Boltzmann law:
SU = εGσTS4 + (1-εG) EDA 
Fifth, the absorbed surface radiation is the upward surface radiation multiplied by the absorptivity of the atmosphere (= emissivity at similar temperatures):
AA = εASU 
So if we put  -> , we get:
ED = εGεAσ(TS – ΔT)4 
And if we put  -> , we get:
AA = εGεAσTS4 + EDεA(1-εG)/εG 
We are almost there. Remember that we wanted to find the ratio ED/AA. Unfortunately, the AA term includes ED and we can’t eliminate it (unless I missed something).
So let’s create the ratio and see what happens. This is equation 6 divided by equation 7 and we can eliminate εA that appears in each term:
ED/AA = [ εGσ(TS - ΔT)4 ] / [ εGσTS4 + ED(1-εG)/εG ] 
And just to make it possibly a little clearer, we will divide top and bottom by εG and color code each part:
ED/AA = [ σ(TS - ΔT)4 ] / [ σTS4 + ED(1-εG)/εG2 ] [8a]
And so the ratio = blackbody radiation at the atmospheric temperature divided by
( blackbody surface radiation plus a factor of downward atmospheric radiation that increases as εG reduces )
We didn’t make a blackbody assumption, it is just that most of the emissivity terms canceled out.
What Does the Maths Mean?
Take a look at the green term – if εG = 1 this term is zero (1-1=0) and the equation simplifies down to:
ED/AA = (TS – ΔT)4 / TS4
Which is very simple. If ΔT = 0 then ED/AA = 1.
Let’s plot ED vs AA for a few different values of ΔT and for TS = 288K:
Compare this with figure 1 (Miskolczi’s fig 2).
Note: I could have just cited the ratios of ED/AA, which – in this graph – are constant for each value of ΔT.
And we can easily see that as ΔT →0, ED/AA →1. This is “obvious” from the maths for people more comfortable with equations.
That’s the simplest stuff out of the way. Now we want to see what happens when εG < 1. This is the interesting part, and when you see the graph, please note that the axes are not the same as figure 4. In figure 4, the graph is of ED vs AA, but now we will plot the ratio of ED/AA as other factors change.
Take a look back at equation 8a. To calculate the ratio we need a value of Ed, which we don’t have. So I use some typical values from Miskolczi – and it’s clear that the value of Ed chosen doesn’t affect the conclusion.
You can see that when εG = 1 the ratio is almost at 0.99. This is the slope of the top line (ΔT=1) in figure 4.
But as surface emissivity reduces, ED/AA reduces
This is clear from equation 8a – as εG reduces below 1, the second term in the denominator of equation 8a increases from zero. As this increases, the ratio must reduce.
In Miskolczi’s graph, as εG changed from 1.0 → 0.96 the calculated ratio increased. I believe this is impossible.
Here is another version with a different value of ΔT:
Perhaps I made a mistake in the maths. It’s pretty simple – and out there in the open, so surely someone can quickly spot the mistake.
Of course I wouldn’t have published the article if I thought it had a mistake..
On conceptual grounds we can see that as the emissivity of the surface reduces, it absorbs less energy from the atmosphere and reflects more radiation back to the atmosphere.
This must reduce the value of ED and increase the value of AA. This reduces the ratio ED/AA.
In Miskolczi’s 2007 paper he shows that as emissivity is reduced from a blackbody to a more realistic value for the surface, the ratio goes in the other direction.
If my equations are correct then the equations of energy balance (for his paper) cannot have been correctly written for the case εG <1.
This one should be simple to clear up.
Update May 31st - Ken Gregory, a Miskolczi supporter appears to agree – and calculates ED/AA=0.94 for a real world surface emissivity.
Other articles in the series
The Mystery of Tau – Miskolczi - introduction to some of the issues around the calculation of optical thickness of the atmosphere, by Miskolczi, from his 2010 paper in E&E
Part Two – Kirchhoff - why Kirchhoff’s law is wrongly invoked, as the author himself later acknowledged, from his 2007 paper
Part Three – Kinetic Energy - why kinetic energy cannot be equated with flux (radiation in W/m²), and how equation 7 is invented out of thin air (with interesting author comment)
Part Five – Equation Soufflé - explaining why the “theory” in the 2007 paper is a complete dog’s breakfast
Part Six – Minor GHG’s - a less important aspect, but demonstrating the change in optical thickness due to the neglected gases N2O, CH4, CFC11 and CFC12.